MPRAMunich Personal RePEc Archive

Optimal Asset Allocation of a PensionFund: Does The Fear of Regret Matter?

Oyakhilome Ibhagui

Baum Tenpers Global Investment Research

19 November 2016

Online at https://mpra.ub.uni-muenchen.de/75802/MPRA Paper No. 75802, posted 25 December 2016 13:08 UTC

Optimal Asset Allocation of a Pension Fund: Does The Fear of Regret Matter?

Oyakhilome Ibhagui1

November, 2016

Abstract In this paper, we incorporate regret into the decision-making process of a pension fund and derive the optimal

asset allocation of a final-wealth-maximizing pension fund in the accumulation and decumulation phases. We

find that the optimal asset allocation must be congruent in both phases if and only if the pension fund is upside

regret averse. In particular, our results suggest that allocation to risky assets must increase through time in the

accumulation and decumulation phases so that the pension fund can realize gains from any upsides in the risky

asset market, thereby maximizing final wealth and limiting the feeling of regret ex-post. Although decisions in

both phases are congruent, we find that the optimal asset allocation generally depends on wealth levels. This

evidence implies that separate management of the accumulation and decumulation phases of a pension fund

decreases available wealth levels and is not an optimal strategy.

Keywords

Financial Markets. Asset allocation. Log-logistic. Modified utility. Mortality. Pension fund. Regret aversion

JEL: G23, G11 and C61

1 Acknowledgements: I am grateful to Diana Barro, School of Advanced Economics, CaFoscari University of Venice, for helpful comments on the earlier version of this paper. The paper also benefited from reviews and helpful suggestions from Akeju Martins. I acknowledge a very generous financial support from the European Union Erasmus Mundus during the period in which the paper was written. All noticeable and hidden omissions, commissions and errors are fully mine. Address for correspondence: Wallace Ibhagui, Department of Economics, D3-06 Keynes College, University of Kent, Canterbury, CT2 7NP, UK. Phone: +00 + 44 (0)1227824319. Fax: +00 +44 (0)1227 827850. Email: oi50@kent.ac.uk

1. Introduction

The major contribution of this paper lies in the use of regret theory to analyze the optimal asset allocation of a

pension fund that aims to maximize the expected modified utility of its final wealth. Unlike in the standard

expected utility framework wherein a pension fund independently considers only the investment choices it makes

and performs expected utility maximization based on these choices, without recourse to other potential choices

that could have been made, regret theory gives room for the pension fund to account for its favoured investment

choices as well as other feasible investment choices that could be made. In essence, the fund pension experiences

regret if the outcome of its investment choices is worse than the outcome of at least one of the forgone investment

choices, and it rejoices if otherwise. Thus, because of the possibility of future regret, the objective function—the

expected modified utility of the pension fund’s final wealth— is set up in such a way as to incorporate a function

that captures feelings of potential regret. This way, we are able to develop a set-up aimed at examining the extent

to which the anticipation of future regret influences current investment choices and optimal asset allocation of a

pension fund. The presence of a regret function distinguishes our framework, the regret theory framework, from

traditional expected utility framework and serves as a novel contribution to the literature..

Accessible literature on asset allocation problems and optimal portfolio strategies for pension funds largely

neglects regret theory and widely favors traditional expected utility maximization. In addition to the many

documented limitations and violations of the traditional expected utility theory so elegantly demonstrated in the

behavioral economics literature, our major discontent with the theory is that it assumes economic agents consider

each outcome independently and disregard other possible outcomes. This can be interpreted to mean that pension

fund managers, for example, care only about the investment choices they make. However, in practice, fund

managers do experience a feeling of regret whenever forgone investment choices yield better returns ex-post than

the choices they made ex-ante. This causes them to incorporate the possibility of regret in their subsequent

investment decisions and asset allocation. As a simple illustration, consider a fund manager who can receive a $5

return on investment for each dollar invested in the debt capital market and either a $7.5 or $1.5 return on

investment for each dollar invested in the equity capital market. If he takes a huge position in the equity capital

market and finally receives a $1.5 return on investment for each dollar invested, he almost surely will experience

a feeling of regret for getting less than he could have gotten had he taken most positions in the debt capital

market. Next time, this experience will shape his investment decisions and therefore make him averse to regret,

and this aversion will induce him to incorporate regret into his decision-making process. Despite this intuitive

perspective on regret, to the best of our knowledge, nowhere has this common human behavioral tendency been

incorporated into the investment decision making process to analyze the optimal asset allocation of a pension

fund. It is therefore in this area that this paper fills a void in the literature.

Generally speaking, the concept of regret, developed by Bell, Loomes and Sugden (1982), is intuitive and

proposes a normative theory of choice under uncertainty that explains many observed violations of the axioms

upon which the traditional expected utility theory is built. Regret is a cognitively mediated emotion of pain and

anger when people observe ex-poste that they took a bad decision ex-ante and could have taken an alternative

decision with a better outcome. In capital markets, people experience regret when their investments give a worse

performance than an alternative investment they could have easily chosen a priori. This, for instance, is in

contrast with disappointment, which is experienced when a negative outcome happens relative to prior

expectations. Regret, which is a powerful negative emotion, is strongly associated with a feeling of responsibility

for a choice made and is known to influence decision-making under uncertainty. It involves the regret/rejoice that

a person feels when he gets outcome 𝑥 instead of outcome 𝑦. The theory assumes people are rational but base

their decisions not only on expected payoffs or utility but also on expected regret, so that they try to anticipate

future regret and consistently incorporate it into their investment decision making process. The incorporation of

regret yields a modified utility and people reach their investment decisions by maximizing the expected value of

this modified utility. This makes regret theory suitable for analyzing the optimal asset allocation of a pension

fund.

The anticipation of future regret is so strong that it forces even Harry Markowitz to relook his very own Nobel

winning asset allocation theory when confronted with a financial decision on his pension plan. His quote: ‘I

should have computed the historical covariance of the asset classes and drawn an efficient frontier. Instead I

visualized my grief if the stock market went way up and I wasn’t in it—or if it went way down and I was

completely in it. My intention was to minimize future regret, so I split my pension scheme contributions 50-50

between bonds and equities.’ ‘Harry Markowitz, as quoted in Zweig (1998), America’s top pension fund’,

Money, 27, page 114. This gives further support and credibility to the claim that regret does influence optimal

investment decision of a pension fund. Anticipation of future experience of regret may lead individuals to make

certain decisions that contrast with expected utility paradigm. This assertion will be investigated in the context of

the optimal asset allocation of a pension fund in this paper.

Meanwhile, unlike other fund managers, the case of pension funds requires the introduction of two

characteristics: (i) the different behaviors of the managed funds during the accumulation (Ac) and decumulation

(Dc) phases, and (ii) mortality risk. In addition, we must take cognizance of regret risk because we aim to work in

a regret theoretic framework. So, this paper considers three dimensions of risk: traditional risk (volatility of final

wealth), regret risk and mortality risk. To the best of our knowledge, no work on optimal asset allocation has

simultaneously considered these risks. The only work, at least to our knowledge, which considers these risks in

asset allocation theory, does not consider them simultaneously. For instance, Bajeux-Besnainou and Jordan

(2001) consider only volatility risk, Michenaud and Solnik (2008) consider volatility risk and regret risk and

Battocchio, Menoncin and Scaillet (2007) consider volatility risk and mortality risk. While the intuition of

applying regret theory to asset allocation is not new, this is the first time that a formal regret theoretic approach is

applied to the optimal asset allocation of a pension fund facing the aforementioned risks.

As we have motivated above, regret is a major factor when making investment choices because institutional

investors, more often than not, care about their choices relative to other strategies they could have employed.

Although evidence favoring the influence of regret on decision-making exists instinctively, it is surprising that

the theory has caught only little attention in the field of finance. In the few available studies, Muermann, Mitchell

and Volkman (2006) apply regret theory to asset allocation in defined contribution pension schemes. They find

that an investor who takes regret into account will hold more risky assets (stocks) when the equity premium is

low but less risky assets when the equity premium is high. Braun and Muermann (2003) apply regret theory to

demand for insurance. Dodonova and Khoroshilov (2005) apply a pseudo regret theory to asset pricing. Heybati,

Rahnamay and Moosavi (2011) apply regret theory to portfolio optimization. Michenaud and Solnik (2008) study

currency hedging techniques for foreign assets in a regret theoretic framework and derive some interesting

implications for long and short hedging positions when a foreign currency appreciates or depreciates ex-post.

However, all these models offer approximate explicit investment rules outside the context of a pension fund.

In this paper, instead, we provide explicit optimal solutions for investment rules within the context of a pension

fund which manages employees’ contributions towards retirement. Our methodology allows the derivation of

approximate closed-form solutions for optimal investment choices available to a pension fund. In particular,

during the active years of the employees, the fund wealth increases because of the contributions that the

employees make towards retirement while, after retirement, the fund wealth decreases because of the pension

payments that the pension fund makes to the retired employees. Following Battocchio, Menoncin and Scaillet

(2007), we suppose that a representative employee has no other choice at the retirement date than to receive a

pension until the death time 𝜏, which is assumed to be a random variable. The pension fund then maximizes the

expected modified utility of its final wealth, in anticipation of future regret. In our model, the contribution and

pension rates are constant and linked by a feasibility condition that guarantees the convenience of both the

pension fund and the representative employee to amicably enter the pension contract. We argue why this

feasibility condition must hold and derive its approximate closed-form expression under the assumption that the

death time 𝜏 follows a log-logistic distribution. We emphasize that our result is quite different from the closed-

form expressions obtained under the assumption of both a Gompertz-Makeham and a Weibull distributed death

time 𝜏 as in Battocchio, Menoncin and Scaillet (2003, 2007). We also remark that the motivation for our choice

of distribution for the death time 𝜏 stems from the fact that death-survival analyses for a random death time are

best done under the assumption of a log-logistic distribution.

Furthermore, we consider that a pension fund does not only manage retirement funds preretirement when

contributions towards future pensions are made, but also manages the remaining wealth postretirement when

pensions are being paid. Therefore, since management of the remaining wealth postretirement is also the duty of

a pension fund, we set up the required optimization problem for the optimal asset allocation during the entire

life of the representative subscriber in such a way that the final date of the optimization problem coincides with

the death time of the subscriber. After solving this optimization problem in a regret theoretic framework, we

find that the optimal portfolio compositions of the pension fund are identical during the accumulation and

decumulation phases. In particular, we show that the amount of wealth invested in the risky-asset class increases

through time during the accumulation phase and also increases through time after the retirement date, during the

decumulation phase. We claim that this behavior is borne out of regret aversion. Regret averse pension funds

would increasingly retain their positions in the risky-asset class for fear of the potential regret of missing out on

any boom or upside in the risky-asset market.

To summarize, in addition to other important results, our major contribution in this paper is systematic. We

integrate regret into a well-defined objective function and this allows us to derive optimal investment strategies

that reflect the risk and regret aversion of a pension fund. We find that the optimal asset allocation must be

congruent in both phases if and only if the pension fund is upside regret averse. In particular, our results suggest

that allocation to risky assets must increase through time in both accumulation and decumulation phases, so that

the pension fund can realize large gains from the upside potential of the risky-asset market, thereby maximizing

final wealth and limiting the feeling of regret ex-post. Although decisions in both phases are congruent, we find

that the optimal asset allocation generally depends on wealth levels. This evidence implies that separate

management of the accumulation and decumulation phases of a pension fund decreases available wealth levels

and is not a robust strategy.

The rest of the paper flows as follows. Section 2 explains some important concepts that will aid the

understanding of other ideas in subsequent sections, presents and discusses the financial model, and elaborates

on the computation of feasibility condition on contribution and pension rates when the death time 𝜏 follows a

log-logistic distribution. Section 3 presents the regret theoretic framework for the pension fund maximization

problem, discusses regret theory and its importance for decision making under uncertainty and describes our

modeling framework. Section 4 develops the objective function, applies regret theory to a pension fund which

seeks to maximize its expected modified utility of final wealth, and computes the optimal allocation rule.

Section 5 discusses the theoretical results in relation to the effective management of a pension fund and

concludes with directions for future research.

2. The Financial Markets Pension Fund Model

We follow Brennan, Schwartz and Lagnado (1997) and Battocchio, Menoncin and Scaillet (2007) and consider

a financial market with one risky-asset class (common stock) and one riskless-asset class (T-bills) having rates

of return 𝑑𝑆𝑆

and 𝑑𝐺𝐺

respectively, and where 𝑟 is the short term interest rate. If we assume also that dividend

yield 𝛿 on common stocks influences returns on risky-asset class, then the joint price process follows

{

𝑑𝑆

𝑆= 𝜇𝑆𝑑𝑡 + 𝜎𝑆𝑑𝑧𝑆, 𝑆(𝑡0) = 𝑆0

𝑑𝐺

𝐺= 𝑟𝑑𝑡, 𝐺(𝑡0) = 𝐺0

𝑑𝑟 = 𝜇𝑟𝑑𝑡 + 𝜎𝑟𝑑𝑧𝑟𝑑𝛿 = 𝜇𝛿𝑑𝑡 + 𝜎𝛿𝑑𝑧𝛿

(1)

where the parameters μi, σi (i = r, δ, S) are at most functions of the variables r, δ, S, and 𝑑𝑧𝑖 are increments due

to Weiner process. 𝑆0 and 𝐺0 are deterministic, positive and represent the initial prices of the risky and riskless-

asset classes respectively, while S and 𝐺 are their prices at time 𝑡 > 0. It should be noted that the dependence of

the return 𝑑𝑆𝑆

on 𝑟 and 𝛿 is completely straightforward. Indeed, 𝑑𝑆𝑆

depends on 𝑟 and 𝛿 through the dependence of

at least one of 𝜇𝑆 and 𝜎𝑆 on at most r, δ, S. This process may be estimated and tested statistically to ascertain the

level of statistical significance of the effect of the state variables on stock returns, but that is not the focus of this

paper.

2.1.1 The Contributions and Pensions Payments

If 𝑈(𝑡) denotes the total amount of contributions to the fund and 𝑉(𝑡) denotes the total amount of pensions paid

by the fund, then 𝑈(𝑡) and 𝑉(𝑡) follow the ordinary linear differential equations

𝑑𝑈(𝑡) = 𝑢𝑑𝑡, 𝑡0 ≤ 𝑡 < 𝑇 (2𝑎)

𝑑𝑉(𝑡) = 𝑣𝑑𝑡, 𝑇 ≤ 𝑡 < 𝜏 (2𝑏)

where 𝑢 > 0 and 𝑣 > 0 are constant and do not vary with time. The pensions are paid until the death time of the

subscriber and do not depend on the investment performance of the fund.

2.1.2 The Feasibility Condition

This is the condition that has to be satisfied before the subscriber and pension fund enter a pension contract in

the first place. For this reason, the pension fund cannot freely dictate the contributions and pensions while the

subscriber cannot solely dictate the pensions. The contributions and pensions cannot be chosen separately. The

subscriber and the pension fund have to reach an agreement on the contributions and pensions simultaneously.

When the subscriber enters the fund, he anticipates that the expected present value of all pensions cannot be

lower than the expected present value of all contributions. Similarly, the pension fund formalizes the contract

with the subscriber when it is convinced that the expected present value of all pensions cannot be more than the

expected present value of all contributions.

Money enters and leaves the pension fund according to the rate

𝑚(𝑡) =𝑑𝑈(𝑡)

𝑑𝑡𝕀𝑡<𝑇 −

𝑑𝑉(𝑡)

𝑑𝑡𝕀𝑡≥𝑇 (3𝑎)

or

𝑚(𝑡) = 𝑢𝕀𝑡<𝑇 − 𝑣(1 − 𝕀𝑡<𝑇),

where

𝕀𝑡<𝑇 = {1, if 𝑡 < 𝑇0, if 𝑡 ≥ 𝑇

If 𝑢 represents the constant contribution rate to the pension fund and 𝑣 represents the constant pension rate paid

to a representative subscriber, then the feasibility condition holds for the pair (𝑢, 𝑣), 𝑢 > 0 and 𝑣 > 0 if

𝔼 [∫𝑚(𝑡)𝑒−𝑟𝑡𝑑𝑡

𝜏

0

] = 0, and the resulting expression for the feasibility condition is

𝑢

𝑣= −1 +

1 − 𝔼[𝑒−𝑟𝜏]

1 − 𝔼[𝑒−𝑟𝜏𝕀𝜏<𝑇] − 𝑒𝑟𝑇ℙ(𝜏 ≥ 𝑇), (3𝑏)

where 𝜏 is the random death time. The proof is in Appendix 1 and the computation of the closed form

approximations for the feasibility condition is in Appendix 2.

3. Pension Fund Maximization Problem in a Regret Theoretic Framework

In this section we approach the problem of a pension fund from a regret/rejoicing point of view. Since pension

funds are largely connected with collecting, pooling and investing funds contributed by sponsors and

beneficiaries in order to provide means for the beneficiaries to accumulate savings over their active working

years so as to finance their consumption needs in retirement, we shall assume that the main problem of a

pension fund is to maximize the expected modified utility of its final wealth at the death time of its subscribers.

Contrary to the traditional choiceless utility function which is defined on outcomes of actual investment choices

that a pension fund makes, the modified utility function does not only consider outcomes of actual investment

choices, but it also includes a comparison of these outcomes with the outcomes of other investment choices that

a pension fund could have made, but has not made. This makes it possible to incorporate regret into the

modified utility function for the purpose of decision making. Regret-conscious pension funds do not only care

about the expected return and volatility of their invested funds, but they also care about the deviations of the

outcomes of their actual choices from the outcomes of their forgone choices. So, they face both volatility risk

and regret risk. Volatility risk is linked to deviations of the invested fund’s return from its expected value.

Regret risk is the risk that the pension funds are going to experience a feeling of regret in the future. That is, the

risk that the outcomes of their actual investment choices will be worse than the outcomes of their forgone

investment choices. There is also mortality risk, which is the risk of death of a subscriber and it enters the

problem through the feasibility condition.

In our work, we suppose that a pension fund can make two choices. The first is to invest/allocate a strictly

positive amount 0 < 𝛼 < 1 to a risky-asset class and the remaining to a riskless-asset class while the second is

to allocate nothing 𝛼 = 0 to the risky-asset class and allocate all to the riskless-asset class. We suppose that the

pension fund is unsure of the performance of the risky-asset class, but always knows beforehand the

performance of the riskless-asset class, so that this performance serves as a benchmark with which the pension

fund compares the outcome of its actual investment strategy. The pension fund is assumed to experience a

feeling of regret or rejoicing on the outcome of its choice. If we suppose that the pension fund makes or prefers

the first choice (of taking a huge position in the risky-asset class) knowing full well that it may later regret its

action or decision if it so happens that the outcome (performance) of the second choice (of taking no position in

the risky-asset class) turns out much better than the outcome of his first choice, how would its optimal

allocation choice/decision be today in order to maximize the expected modified utility of its final wealth,

improve its feeling of rejoicing and lessen its feeling of regret that may occur in the future/ex-post after it has

exited it position to evaluate its proceeds? This is one of the questions we shall answer in this part of our

research.

3.1.1 The Maximization Problem Setup

As we have motivated in the previous section, regret theory rests on two fundamental assumptions. The

first is that agents experience the sensations of regret and rejoicing, and the second is that agents try to

anticipate and take account of these ex-post sensations when making ex-ante decisions under

uncertainty. The modified utility function is therefore defined over the ex-post (final) outcomes of

choices and rational investors would make choices ex-ante by maximizing the expected value of this

modified utility. This allows agents to take the anticipation of regret into account in an axiomatic

fashion. The modified utility function is not only defined over the outcome of the choice an agent

makes, but it also includes a comparison with the outcome of another choice that could have been made

in the same state of the world.

If we define the expected value of the modified utility of a rational regret averse agent faced with two

choices by

𝔼(𝜓(𝑥, 𝑦)) = 𝔼 (𝑈(𝑥) + 𝑓(𝑈(𝑥) − 𝑈(𝑦))) (4𝑎)

then the agent will seek to make a choice ex-ante that will give a final outcome 𝑥 which will maximize

his expected modified utility, i.e.

max 𝑥

𝔼(𝜓(𝑥, 𝑦)) = ∇ (4b)

with ∇= max𝑥𝔼(𝑈(𝑥) + 𝑓(𝑈(𝑥) − 𝑈(𝑦))).

It is important to note that if agents experience no regret or rejoicing at all, or if the function 𝑓 is linear,

then the above formulation collapses to the conventional expected utility paradigm.

Now that we have discussed the relevant aspects of regret theory that will aid our work on pension

funds, we will proceed to how we can apply it in a pension fund context. Therefore, in what follows, we

shall study the investment behavior of a pension fund in a regret theoretic framework.

3.1.2 The Managed Wealth of the Pension Fund

In the preceding section, we stated that the pension fund is assumed to have two choices—either it

invests a strictly positive amount in the risky-asset class or it invests nothing in the risky-asset class.

We further suppose that the pension fund does not prefer the choice of taking no position in the risky-

asset class. This is the choice or decision of the pension fund. If 𝜃(𝑡) represents the number of units of

the risky-asset class and ∅(𝑡) represents the number of units of the riskless-asset class, then the

apportioned wealth to the risky-asset class is 𝜃(𝑡)𝑆(𝑡) and the allotted wealth to the riskless-asst class

is ∅(𝑡)𝐺(𝑡). Therefore, the total wealth process 𝑊(𝑡) of the pension fund, which we shall call the

outcome of its choice of taking a positive position in the risky-asset class, evolves according to

𝑊(𝑡) = 𝜃(𝑡)𝑆(𝑡) + ∅(𝑡)𝐺(𝑡) (5𝑎)

The associated stochastic differential equation is

𝑑𝑊 = 𝜃𝑑𝑆 + ∅𝑑𝐺 + (𝑆 + 𝑑𝑆)𝑑𝜃 + 𝐺𝑑∅ (5𝑏)

Introducing the self-financing condition suggests that changes in portfolio composition are only due to

variations in the prices of assets constituting the portfolio. Therefore, the term (𝑆 + 𝑑𝑆)𝑑𝜃 + 𝐺𝑑∅ should

normally be nonexistent or zero. However, since we are considering the case of a pension fund, then, without

any loss of generality, we can argue that the self financing condition ensures that the additional term (𝑆 +

𝑑𝑆)𝑑𝜃 + 𝐺𝑑∅ comes from the contributions 𝑢 made by the subscriber during the accumulation phase and it is

used to finance the pension payments 𝑣 during the decumulation period. Accordingly, therefore,

(𝑆 + 𝑑𝑆)𝑑𝜃 + 𝐺𝑑∅ = 𝑚𝑑𝑡 (6𝑎)

where 𝑚 is the rate at which money enters and leaves the fund, and so

𝑑𝑊 = (𝑊𝑟 + ɸ(𝜇𝑆 − 𝑟) +𝑚)𝑑𝑡 + ɸ𝜎𝑆𝑑𝑧𝑆 (6𝑏)

where ɸ = 𝜃𝑆 = 𝛼𝑊 is the amount of wealth apportioned to the risky-asset class, 𝑑𝑆 = 𝑆(𝜇𝑆𝑑𝑡 + 𝜎𝑆𝑑𝑧𝑆) and

𝑑𝐺 = 𝐺𝑟𝑑𝑡 represent the price process for the risky-asset class and the riskless-asset class respectively.

Now, since we want to perform the analysis of a pension fund in a regret theoretic framework, we will not only

consider the outcome of the decision/choice made by the pension fund, but we shall also consider the outcome

of another choice that the pension fund could have made. In particular, we consider a case in which the pension

fund could have made a choice of taking no position in the risky-asset class, i.e. investing solely in the riskless-

asset class. The wealth process or outcome of such a choice/decision would be

𝑊𝑜(𝑡) = ∅(𝑡)𝐺(𝑡) (7𝑎)

and its associated stochastic differential equation would be

𝑑𝑊𝑜 = ∅𝑑𝐺 + 𝐺𝑑∅ (7𝑏)

Self-financing condition would then imply that

𝐺𝑑∅ = 𝑚𝑑𝑡 (8𝑎)

where 𝑚 is as before, and so

𝑑𝑊𝑜 = (𝑊𝑜𝑟 + 𝑚)𝑑𝑡 (8𝑏)

Equations (5a) –8(b) explicitly show the outcome of the actual investment choice made by the pension fund as

well as the outcome of a forgone investment choice that the pension fund could have made. In what follows, we

will now proceed and write down the objective function of the pension fund in a regret theoretic framework.

4 The Objective Function of the Pension Fund

The objective of the pension fund is to maximize the expected modified utility of its final wealth/final outcome

at the death time 𝜏 of its subscriber, under the assumption that the pension fund anticipates to experience a

feeling of regret if the outcome of its investment choice is less than the outcome of a forgone investment choice

and that the pension fund takes this feeling into account when making its decision under uncertainty with the

sole aim of curtailing future regrets. The forgone investment choice is taken as a full investment in the riskless-

asset class. Following (4a), we can write the modified utility function of the pension fund as

𝜓(𝑊,𝑊𝑜) = (𝑈(𝑊)+ 𝑓(𝑈(𝑊) − 𝑈(𝑊𝑜)))

= (𝑈(𝜃𝑆 + ∅𝐺) + 𝑓 (𝑈(�̂�𝑆 + ∅𝐺) − 𝑈(∅𝐺))) (9𝑎)

Thus, the objective function of the pension fund is

max ɸ

𝔼𝑡0𝜏 ( 𝜓(𝑊(𝜏),𝑊𝑜(𝜏))) = ∇ (9b)

where

∇= maxɸ𝔼𝑡0𝜏 ( 𝑈(𝜃𝑆 + ∅𝐺) + 𝑓 (𝑈(�̂�𝑆 + ∅𝐺) − 𝑈(∅𝐺)))

i.e. finding the right allocation to the risky-asset class that will help maximize the expected modified utility of

the pension fund’s final wealth. Following [4] and [9], we assume the death time 𝜏 is independent of all other

sources of risk. With this assumption, we can rewrite the above expected value as

𝔼𝑡0𝜏 ( 𝜓(𝑊(𝜏),𝑊𝑜(𝜏))) = 𝔼𝑡0 [∫ 𝑔(𝑡)𝜓(𝑊(𝑡),𝑊

𝑜(𝑡))𝑑𝑡

∞

𝑡0

] (10)

where 𝑔(𝑡) = 𝑛(𝑡)𝑝𝑡0𝑒−𝜌(𝑡−𝑡0) is the actuarial discount factor, 𝑛(𝑡) is the instantaneous mortality rate (known

as mortality force), 𝑝𝑡0 is the survival probability and 𝜌 is the positive intertemporal discount rate [4].

Now, we need to define the utility function 𝑈(∗) and the regret function 𝑓(∗) that we shall use in the course of

our analysis. The one most widely used utility function in the literature is the constant relative risk aversion

utility function of the form 𝑈(𝑋) = 𝑋1−𝜗

1−𝜗 with 1 − 𝜗 < 1. Here, we shall use a slight modification of this utility

function. Now, we know that a pension function derives utility from its wealth after all contributions have been

made and pensions have been paid. Therefore, if we let 𝑀(𝑡) represent all contributions and payments up to the

present time, where 𝑀(𝑡) can be written as [4]

𝑀(𝑡) = ∫𝑚(𝑠)𝑒−𝑟(𝑠−𝑡)𝑑𝑠

𝑡

𝑡0

then the pension fund will derive some utility from 𝑊(𝑡) −𝑀(𝑡), and so the utility function must be defined on

this argument, i.e. 𝑈(𝑊(𝑡) − 𝑀(𝑡)). This is our slight modification of the utility function. For the regret

function 𝑓(∗), we assume that it is of the form 𝑓(𝑌) = (𝑌 + 𝑎)𝜌, with 𝑎 > 0 and 𝜌 > 0, both of which are

positive and less than 1 [1]. Accordingly, therefore, the modified utility function can be written as

𝜓(𝑊,𝑊𝑜) =(𝑊(𝑡) − 𝑀(𝑡))

1−𝜗

1 − 𝜗+ (

1

1 − 𝜗((𝑊(𝑡) −𝑀(𝑡))

1−𝜗− (𝑊𝑜(𝑡) −𝑀(𝑡))

1−𝜗) + 𝑎)

𝜌

(11)

where all variables are as previously defined.

4.1 The Optimization Problem of the Pension Fund

The previous sections set the ground for formulating the optimization problem of the pension fund. As we

assume that the pension fund seeks to maximize the expected modified utility of its terminal wealth after all

contributions have been made and pensions have been made, we write the regret theoretic asset allocation

problem as

{

max

ɸ𝔼𝑡0 [∫ 𝑔(𝑡) (

(𝑊(𝑡) −𝑀(𝑡))1−𝜗

1 − 𝜗+ (

1

1 − 𝜗((𝑊(𝑡) − 𝑀(𝑡))

1−𝜗− (𝑊𝑜(𝑡) −𝑀(𝑡))

1−𝜗) + 𝑎)

𝜌

) 𝑑𝑡

∞

𝑡0

]

𝑑ɸ = (𝑊𝑟 + ɸ(𝜇𝑆 − 𝑟) +𝑚)𝑑𝑡 + ɸ𝜎𝑆𝑑𝑧𝑆 𝑑𝑊𝑜 = (𝑊𝑜𝑟 + 𝑚)𝑑𝑡

𝑊(𝑡0) = 𝑊0

𝑊𝑜(𝑡0) = 𝑊0𝑜

(12)

where 𝑔(𝑡) is the actuarial discount factor [4].

Following [4], let us define the function

{𝑉(𝑡,𝑊,𝑊0,ɸ) = [∫𝑔(𝑠) (

(𝑊(𝑠) −𝑀(𝑠))1−𝜗

1 − 𝜗+ (

1

1 − 𝜗((𝑊(𝑠) −𝑀(𝑠))

1−𝜗− (𝑊𝑜(𝑠) −𝑀(𝑠))

1−𝜗) + 𝑎)

𝜌

)𝑑𝑠

∞

𝑡

]

.

then the value function can be written as

𝐽(𝑡,𝑊,𝑊0) = maxɸ 𝑉(𝑡,𝑊,𝑊0, ɸ)

This value function satisfies the HJB Hamilton-Jacobi-Bellman equation

𝐽𝑡 +maxɸ

[

{

𝑔(𝑡) (

(𝑊(𝑡) −𝑀(𝑡))1−𝜗

1 − 𝜗+ (

1

1 − 𝜗((𝑊(𝑡) − 𝑀(𝑡))

1−𝜗− (𝑊𝑜(𝑡) − 𝑀(𝑡))

1−𝜗) + 𝑎)

𝜌

)

+

𝐽𝑊(𝑊𝑟 + ɸ(𝜇𝑆 − 𝑟) +𝑚) +1

2𝐽𝑊𝑊ɸ2𝜎𝑆2 + 𝐽𝑊0(𝑊𝑜𝑟 + 𝑚) ]

= 0

From this equation, the first order condition for the maximization yields

[

{

𝑔(𝑡)(𝑊(𝑡) −𝑀(𝑡))

−𝜗(1 + 𝜌 (

1

1 − 𝜗((𝑊(𝑡) − 𝑀(𝑡))

1−𝜗− (𝑊𝑜(𝑡) − 𝑀(𝑡))

1−𝜗) + 𝑎)

𝜌−1

)

+𝐽𝑊𝜇𝑆 + 𝐽𝑊𝑊𝜎𝑆

2ɸ ]

= 0

⇒ ɸ∗ = −ℙ

𝐽𝑊𝑊𝜎𝑆2 (13)

where the subscripts indicate the partial derivatives with respect to 𝐽 and

ℙ =

{

𝑔(𝑡)(𝑊(𝑡) −𝑀(𝑡))−𝜗(1 + 𝜌 (

1

1 − 𝜗((𝑊(𝑡) − 𝑀(𝑡))

1−𝜗− (𝑊𝑜(𝑡) −𝑀(𝑡))

1−𝜗) + 𝑎)

𝜌−1

)

+𝐽𝑊𝜇𝑆

The first order conditions are necessary and sufficient for optimality because the modified utility is concave in

𝑊 and hence in ɸ, since 𝑊 is a function of ɸ. In fact, under all the suitably stated conditions that must hold for

the optimization problem of the pension fund, the modified utility function, and hence the value function, is

increasing and concave in 𝑊. The proof is given in Appendix 3.

After substituting the expression of ɸ∗, we obtain the resulting Hamilton-Jacobi-Bellman equation

[

{

𝐽𝑡 + 𝑔(𝑡) (

(𝑊(𝑡) −𝑀(𝑡))1−𝜗

1 − 𝜗+ (

1

1 − 𝜗((𝑊(𝑡)−𝑀(𝑡))

1−𝜗− (𝑊𝑜(𝑡)−𝑀(𝑡))

1−𝜗) + 𝑎)

𝜌

)

+

𝐽𝑊(𝑊𝑟 +𝑚) +

ℵ

𝐽𝑊𝑊𝜎𝑆2 (1

2ℵ − 𝐽𝑊(𝜇𝑆 − 𝑟)) + 𝐽𝑊0(𝑊𝑜𝑟 +𝑚)

]

= 0

For the value function, we can try the substitution 𝐽(𝑡,𝑊,𝑊0) = ℎ(𝑡)𝑔(𝑡) 𝜓(𝑊,𝑊𝑜), where ℎ(𝑡) is a function

that needs to be determined [4]. From this substitution, we have

𝐽𝑡 = 𝑔(𝑡)𝜓(𝑊,𝑊𝑜) 𝜕ℎ(𝑡)

𝜕𝑡+ ℎ(𝑡)𝜓(𝑊,𝑊𝑜)

𝜕𝑔(𝑡)

𝜕𝑡+ ℎ(𝑡)𝑔(𝑡)

𝜕𝜓(𝑊,𝑊𝑜)

𝜕𝑡

𝐽𝑊 = ℎ(𝑡)𝑔(𝑡)𝜕𝜓(𝑊,𝑊𝑜)

𝜕𝑊, 𝐽𝑊𝑊 = ℎ(𝑡)𝑔(𝑡)

𝜕

𝜕𝑊

𝜕𝜓(𝑊,𝑊𝑜)

𝜕𝑊, 𝐽𝑊𝑜 = ℎ(𝑡)𝑔(𝑡)

𝜕𝜓(𝑊,𝑊𝑜)

𝜕𝑊𝑜

Plugging these expressions into the Hamilton Jacobi Bellman equation and simplifying terms, we obtain that

ℎ(𝑡) satisfies

𝜕ℎ(𝑡)

𝜕𝑡+ (

1

𝑔(𝑡)

𝜕𝑔(𝑡)

𝜕𝑡+ 𝑟 (1 +

1

2𝜎𝑆2

𝑟

(1 − 𝜗)(𝜌 − 1)𝑔(𝑡)) +

1

2𝜎𝑆2

𝑔(𝑡)

ℎ(𝑡)

1

(𝜌 − 1)) ℎ(𝑡) + 𝐴(𝑊,𝑊𝑜) = 0

where

𝐴(𝑊,𝑊𝑜) =1

2

ℵ2

𝐽𝑊𝑊𝜎𝑆2 −

𝐽𝑊𝐽𝑊𝑊

ℵ(𝜇𝑆 − 𝑟)

𝜎𝑆2

The precise form of the function ℎ(𝑡) is not important for computing the optimal portfolio composition of the

pension fund since the Arrow-Pratt risk aversion index computed on 𝐽(𝑊,𝑊𝑜) does not depend on ℎ(𝑡). This

makes it possible for us to obtain the composition of the optimal asset allocation of the pension fund. Thus,

given a pair (𝑢, 𝑣) of constant contribution and pension rates satisfying the feasibility condition, the

composition of the optimal asset allocation of a pension fund which maximizes the expected modified utility of

its final wealth is given by

ɸ∗ =𝐸

(𝐿 − 𝐻)𝜎𝑆2 (14)

where

{

𝐸 = (

1

𝑔(𝑡)+ 𝜇𝑆) [1 +

𝜌(𝜌−1)

1−𝜗(𝑊1−𝜗 −𝑊𝑜1−𝜗) + 𝜌(𝜌 − 1)(𝑊𝑜−𝜗 −𝑊−𝜗)𝑀 + 𝑎𝜌(𝜌 − 1)]

𝐿 = 𝜗 [−1 +𝜌(𝜌−1)

1−𝜗(𝑊1−𝜗 −𝑊𝑜1−𝜗) + 𝜌(𝜌 − 1)(𝑊𝑜−𝜗 −𝑊−𝜗)𝑀 + 𝑎𝜌(𝜌 − 1)]

𝐻 = 𝜌(𝜌 − 1) (1

𝑊𝜗 +𝑀𝜗

𝑊𝜗+1) [1 +𝜌(𝜌−2)

1−𝜗(𝑊1−𝜗 −𝑊𝑜1−𝜗) + 𝜌(𝜌 − 2)(𝑊𝑜−𝜗 −𝑊−𝜗)𝑀 + 𝑎𝜌(𝜌 − 2)]

Without any loss of generality, and for simplicity, we can disregard the last term (set 𝑎 to zero) in each of the above expressions, replace 𝑀 with its value and then break each expression into two parts to get

{

𝐸𝑢 = (1

𝑔(𝑡)+ 𝜇𝑆) [1 +

𝜌(𝜌 − 1)

1 − 𝜗(𝑊1−𝜗 −𝑊𝑜1−𝜗) + 𝜌(𝜌 − 1)(𝑊𝑜−𝜗 −𝑊−𝜗)∫𝑢

𝑡

0

𝕀𝑡<𝑇𝑒−𝑟(𝑠−𝑡)𝑑𝑠]

𝐿𝑢 = 𝜗 [−1 +𝜌(𝜌 − 1)

1 − 𝜗(𝑊1−𝜗 −𝑊𝑜1−𝜗) + 𝜌(𝜌 − 1)(𝑊𝑜−𝜗 −𝑊−𝜗)∫𝑢

𝑡

0

𝕀𝑡<𝑇𝑒−𝑟(𝑠−𝑡)𝑑𝑠]

𝐻𝑢 = 𝜌(𝜌 − 1)(1

𝑊𝜗+

𝜗

𝑊𝜗+1∫𝑢

𝑡

0

𝕀𝑡<𝑇𝑒−𝑟(𝑠−𝑡)𝑑𝑠)

[ 1 +

𝜌(𝜌 − 2)

1 − 𝜗(𝑊1−𝜗 −𝑊𝑜1−𝜗) +

𝜌(𝜌 − 2)(𝑊𝑜−𝜗 −𝑊−𝜗)(∫𝑢

𝑡

0

𝕀𝑡<𝑇𝑒−𝑟(𝑠−𝑡)𝑑𝑠)

]

ɸ𝑢∗ =

𝐸𝑢(𝐿 − 𝐻)𝜎𝑆

2 (14𝑎)

and

{

𝐸𝑣 = (1

𝑔(𝑡)+ 𝜇𝑆) 𝜌(1 − 𝜌)(𝑊

𝑜−𝜗 −𝑊−𝜗)∫𝑣(1 −

𝑡

0

𝕀𝑡<𝑇)𝑒−𝑟(𝑠−𝑡)𝑑𝑠

𝐿𝑣 = 𝜗𝜌(1 − 𝜌)(𝑊𝑜−𝜗 −𝑊−𝜗)∫𝑣(1 −

𝑡

0

𝕀𝑡<𝑇)𝑒−𝑟(𝑠−𝑡)𝑑𝑠

𝐻𝑣 = 𝜌2(1 − 𝜌)(𝜌 − 2)

𝜗

𝑊𝜗+1(∫𝑣(1 −

𝑡

0

𝕀𝑡<𝑇)𝑒−𝑟(𝑠−𝑡)𝑑𝑠)

2

(𝑊𝑜−𝜗 −𝑊−𝜗)

ɸ𝑣∗ =

𝐸𝑣(𝐿 − 𝐻)𝜎𝑆

2 (14𝑏)

so that

ɸ∗ = ɸ𝑢∗ + ɸ𝑣

∗

where 𝑢 and 𝑣 are linked by the feasibility condition and 𝐿 = 𝐿𝑢 + 𝐿𝑣 and 𝐻 = 𝐻𝑢 +𝐻𝑣.

The first set of equations depends explicitly on the contribution rate 𝑢 and the wealth level of the pension fund

as a result of its actual investment choice, i.e. investing a positive amount in the risky- asset class. It also

depends on the wealth level of a forgone investment choice, i.e. the investment choice that could have been

made, which would involve taking no position in the risky-asset class. The wealth level of the forgone

investment decision is a benchmark with which to compare the outcome or wealth level of the pension fund’s

actual investment choice. The second set of equations also depends on the two wealth levels as well as on the

pension rate 𝑣. Furthermore, the mortality risk 𝜏 enters the maximization problem through the link that exists

between the contribution rate 𝑢 and the pension rate 𝑣 in the feasible condition derived in Appendix 1. It is

very pertinent to further stress that 𝑢 and 𝑣 must satisfy this feasibility condition as we have already

demonstrated. As noted in [9], if this link between 𝑢 and 𝑣 is completely not considered through the feasibility

condition, then the composition of the optimal asset allocation of a pension fund is independent of mortality

risk. Such an optimal asset allocation strategy for a pension fund can produce very shallow and extremely

restricted results in practice.

A very important deduction from the two sets of equations is that the optimal allocation of the pension fund

does explicitly depend on the wealth levels. This becomes evident since the equations themselves depend on the

wealth levels and when we substitute the expressions of the equations into the composition of the optimal asset

allocation of the pension fund, we get that it also depends on the wealth levels. From this, therefore, we can

deduce that it is suboptimal for a pension fund to manage the accumulation phase, when contributions are made,

and the decumulation phase, when pensions are paid, separately. Hence our model prohibits the pension fund

from outsourcing any of the phases of the pension fund management to a second or third party. This means that

the idea of outsourcing, commonly employed by large pension funds in emerging economies, is not effective

and does not conform to an optimal strategy in a regret theoretic framework. Ideally, a firm should commit the

entire management of its employees’ retirement plan to the same pension fund and the pension fund itself must

not outsource any of the phases of the pension management process. If it does, its action will be suboptimal and

may have undesirable consequences. We now investigate what happens at the two phases involved in the

management of a pension fund – accumulation and decumulation phases.

4.1.1 Accumulation Phase

When 𝑡 ≤ 𝑇, the pension fund is in the accumulation phase and the optimal allocation strategy in this phase can

be written as

𝐸𝑢 = (1

𝑔(𝑡)+ 𝜇𝑆)(1 +

𝜌(𝜌 − 1)

1 − 𝜗(𝑊1−𝜗 −𝑊𝑜1−𝜗) + 𝜌(𝜌 − 1)(𝑊𝑜−𝜗 −𝑊−𝜗)

𝑢

𝑟(𝑒𝑟𝑡 − 1))

𝐿𝑢 = 𝜗 (−1 +𝜌(𝜌 − 1)

1 − 𝜗(𝑊1−𝜗 −𝑊𝑜1−𝜗) + 𝜌(𝜌 − 1)(𝑊𝑜−𝜗 −𝑊−𝜗)

𝑢

𝑟(𝑒𝑟𝑡 − 1))

𝐻𝑢 = 𝜌(𝜌 − 1) (1

𝑊𝜗+𝑢

𝑟

𝜗

𝑊𝜗+1(𝑒𝑟𝑡 − 1))(1 +

𝜌(𝜌 − 2)

1 − 𝜗(𝑊1−𝜗 −𝑊𝑜1−𝜗) + 𝜌(𝜌 − 2)(𝑊𝑜−𝜗 −𝑊−𝜗)

𝑢

𝑟(𝑒𝑟𝑡 − 1))

ɸ𝑢∗ =

𝐸𝑢(𝐿 − 𝐻)𝜎𝑆

2 and ɸ𝑣∗ = 0 since 𝑡 ≤ 𝑇 ⇒ (∫𝑣(1 −

𝑡

0

𝕀𝑡<𝑇)𝑒−𝑟(𝑠−𝑡)𝑑𝑠) = 0

We first recall that 𝜇𝑆 can be positive, negative or zero and ( 1

𝑔(𝑡)+ 𝜇𝑆) >0 since 𝑔 is a discount factor whose

value is positive and less than 1. Suppose now that the pension fund is not regret averse. During the

accumulation phase, and in the absence of regret aversion, i.e. 𝜌=0, the optimal allocation to the risky-asset

class assumes negative values and thus contains a decreasing proportion of the risky-asset class with respect to

time. This means that, in the absence of regret aversion, the optimal allocation to risky assets during the

accumulation phase decreases through time in order for the pension fund to meet payments of future pensions to

its subscribers. This property is not difficult to check. For instance, when we set the regret coefficient to zero

and differentiate the resulting expression with respect to time, we obtain

𝑑ɸ𝑢∗

𝑑𝑡=1

𝑔2𝑑𝑔

𝑑𝑡< 0 ∀

𝑑𝑔

𝑑𝑡< 0

as previously noted. The graphical illustration is shown as below

Figure 4: Behavior of the optimal asset allocation with time

This behavior during the accumulation phase substantiates the implications of the results documented in

Battocio, Menoncin and Scaillet (2003, 2007). However, in the presence of regret, as in our case, the conclusion

is not always the same. Indeed, we notice that, depending on the level of aversion and the choice of the

underlying variables, the optimal asset allocation to the risky-asset class takes positive values and thus contains

a non-decreasing proportion of the risky-asset class with regard to time. This means that, in the presence of

regret aversion, regret averse pension funds have an optimal risky-asset allocation strategy that increases

through time during the accumulation phase. In our view, the intuition behind this is pragmatically clear. Regret

averse pension funds invest more in risky assets as time passes so as not to miss the upside potential of the

risky-asset market, especially in bullish times. Thus a regret averse pension fund invests an increasing amount

of wealth in the risky asset class. This is done in order to have a higher return on the managed wealth and on the

contributions made by the subscribers.

ɸ*

time, t

4.1.2 Decumulation Phase

When 𝑡 > 𝑇, the pension fund is in the decumulation phase and the optimal asset allocation strategy in this

phase can be written as

𝐸𝑣 =𝑣

𝑟(1

𝑔(𝑡)+ 𝜇𝑆) 𝜌(1 − 𝜌)(𝑊

𝑜−𝜗 −𝑊−𝜗)(𝑒𝑟(𝑡−𝑇) − 1)

𝐿𝑣 =𝑣

𝑟𝜗𝜌(1 − 𝜌)(𝑊𝑜−𝜗 −𝑊−𝜗)(𝑒𝑟(𝑡−𝑇) − 1)

𝐻𝑣 = (𝑣

𝑟)2

𝜌(1 − 𝜌)𝜗

𝑊𝜗+1(𝑒𝑟(𝑡−𝑇) − 1) (𝜌(𝜌 − 2)(𝑊𝑜−𝜗 −𝑊−𝜗)(𝑒𝑟(𝑡−𝑇) − 1))

ɸ𝑣∗ =

(1𝑔(𝑡)

+ 𝜇𝑆)

(𝜗 −𝑣𝑟 𝜌(𝜌 − 2)

𝜗𝑊𝜗+1

(𝑒𝑟(𝑡−𝑇) − 1))𝜎𝑆2

During the decumulation phase, and in the absence of regret aversion, i.e. 𝜌=0, the optimal allocation to the

risky-asset class takes positive values and therefore increases through time when the actuarial discount factor

decreases with time. This property is very easy to check. Indeed, if we set the coefficient of regret aversion to

zero, we obtain

ɸ𝑣∗ =

(1𝑔(𝑡)

+ 𝜇𝑆)

𝜗> 0

which shows that the risky-asset class takes positive values. Furthermore, if we take the time derivative of this

expression, we see that

𝑑ɸ𝑣∗

𝑑𝑡=−1

𝑔2𝑑𝑔

𝑑𝑡> 0 ∀

𝑑𝑔

𝑑𝑡< 0

which shows that the optimal allocation to the risky-asset class increases through time. Again, this conforms to

the deductions made in Battocio, Menoncin and Scaillet (2003, 2007). This behavior is depicted below.

Figure 5: Behavior of the optimal asset allocation with time

ɸ

*

time, t

In the presence of regret, as in our case, the situation is somewhat similar. Indeed, for all admissible levels of

regret aversion, the optimal allocation to the risky-asset class takes only positive values during the decumulation

phase and increases through time. The optimal allocation is skewed in favor of the risky-asset class in the

decumulation phase. In fact when the retirement date 𝑇 is still far away, the pension fund can afford to invest in

the risky-asset class because of the belief that the risky-asset class may provide a better opportunity for it to

accumulate more wealth before the retirement date. This behavior is borne out of regret aversion. The pension

fund is averse to regret because it does not want to experience the feeling of regret that comes when the risky-

asset appreciates over time and the pension fund does not have a high position in it. We recall from the

formulation of our problem that the pension fund feels regret or rejoicing with respect to the risky-asset class.

Therefore, if the pension fund anticipates an ex-post feeling of regret and factors this feeling into its decision

making process, then to maximize its portfolio value, increase its final wealth level and minimize its future

regret the pension fund’s optimal strategy would be to take an increasing position in the risky-asst class during

both the accumulation and decumulation phases.

We must emphasize that this result contrasts the behavior of a pension fund in the traditional expected utility

(EU) framework. In the EU framework, the optimal allocation to risky assets decreases through time in the

accumulation phase so that the pension fund would be able to make a sure and substantial pension payment in

the decumulation phase, while the optimal asset allocation to risky assets increases through time in the

decumulation phase. In our case, however, the allocation to the risky-asset class soars with time during both

phases. In particular, this is so in the decumulation phase because after retirement and during the payment of

pensions, the higher the number of pension installments paid, the fewer remaining pensions left to be paid, and

as the death time approaches, the probability that the pension subscriber would die increases, so the pension

fund can accept to take much risk with less feeling of regret ex-post. The pension fund can afford to invest more

and more in the risky-asset class in order to have a high return on the received contributions from the

subscribers (which are then reinvested for the purpose of growing the contributed funds) and on the total

managed wealth. This behavior is borne out of regret. Regret aversion forces the pension fund to behave this

way so as to minimize the feeling of regret that would occur if the risky-asset goes up and the pension fund is

not there.

When the pension fund starts paying pensions, the higher the number of pension installments paid, the lower the

probability to pay another pension installment since the death probability increases through time. Thus, after the

retirement date 𝑇 when 𝑡 increases, the pension fund can afford to invest increasingly in the risky-asset class

because it has fewer pensions to pay and thus would deem it optimal to increase the allocation to risky assets

because of the upside potential they are capable of generating. Thus, for a pension fund which seeks to

maximize the expected modified utility of its final wealth, the optimal asset allocation rule is such that the

amount allocated to risky-asset class increases through time during both phases— the accumulation phase and

the decumulation phase.

5 Conclusion

Battocchio et al (2007) use the theory of expected utility-maximization, applied to the management of two

phases of a pension fund, the accumulation and decumulation phases, to conclude that the optimal asset

allocation to risky assets in both phases must be different—it must decrease through time in the accumulation

phase and increase through time in the decumulation phase. Instead, in this paper, we have considered the

problem of finding the optimal asset allocation of a pension fund in a regret theoretic framework. We

incorporate regret into the decision making process of a pension fund and thus study the same problem in a

regret theoretic framework. We focused on the composition of the optimal asset allocation strategy in the

accumulation and decumulation phases of the pension fund management. The configuration of the financial

market is such that there is a risky-asset class whose price follows a geometric Brownian motion, a riskless-

asset class paying a non trivial interest rate and the market is not necessarily complete. Furthermore, the pension

fund is assumed to possess a power utility function and a regret/rejoice function which satisfies all the usual

properties of a standard regret function. Our emphasis is on regret theory and we derive approximated closed

form solutions for the pension fund management. We analyzed the optimal allocation problem during both the

accumulation phase and the decumulation phase. We considered a random death time of the representative

subscriber and assumed the death time follows a Log-logistic distribution. We then derive the feasibility

condition connecting the contribution rate 𝑢, the pension rate 𝑣, and the random death time 𝜏 under the

assumption that the contribution and pension rates are constant. This is a major area of contribution in this

paper. There are three risk attributes entering our objective function. The first is the traditional volatility, the

second is the regret risk, embedded in the regret function, and the third is the mortality risk that comes in

through the feasibility condition.

We showed that the optimal asset allocation for a regret averse pension fund in the accumulation phase is not

different from the one in the decumulation phase. This is another major area of contribution in this paper. This

particular result contrasts sharply with that obtained in the expected utility framework in which the optimal

asset allocation during the accumulation phase is completely different from the optimal asset allocation in the

decumulation phase. In expected utility framework particularly, the allocation to risky assets decreases with

time during the accumulation phase so as to make it possible for the pension fund to guarantee the payment of

pensions to the representative subscriber after retirement while, during the decumulation phase when pensions

are paid, the optimal allocation to risky assets increases with time. Instead, in our case, in the regret theory

framework, the optimal allocation to risky-asset class increases during both accumulation and decumulation

phases. The intuition to support this behavior is that when pensions are paid, the probability of paying more

pensions decreases as time passes because of the subscriber’s inevitable closeness to death after the retirement

date. This makes it possible for the pension fund to invest the remaining available wealth in more and more

risky assets and so the allocation to risky assets increases with time. The pension fund is confident enough to

take higher positions in risky assets because of its reducing level of obligations and its desire to benefit from the

upside potential of the risky-asset market. We also observe that the optimal asset allocation of the pension fund

jointly depends on its global wealth levels in both the accumulation phase and the decumulation phase.

Therefore, it is not optimal for a pension fund to manage the accumulation phase and decumulation phase

separately. Outsourcing the management of just a phase of a pension fund is therefore not an optimal strategy. A

pension fund that manages the accumulation phase should ideally manage the decumulation phase.

Future research would be an extension of our model to cater to the optimal asset allocation rule of a pension

fund that manages the retirement proceeds of immortal subscribers in a regret theoretic framework. Although

humans are not immortal and so such an extension would be purely theoretical with very little practical

implications and real life consequences, we believe the outcome would be inspiring to those theoretical analysts

and researchers who are interested in immortality bias in survival analysis as regards the management of

pension funds.

6 References

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[2] Bajeux- Besnainou, I., Jordan, J.V., 2001. Mean-Variance Asset Allocation for Long Horizons. Department of Finance and National Economic Research Associates, 2023/1255. Outstanding Paper in Investments, pages 1-27

[3] Michenaud, S., Solnik, B., 2008. Apply Regret Theory to Investment Choices: Currency Hedging Decisions. Research Working Paper Series, JEL Classification: G11, G15, pages 1-33

[4] Battochio, P., Menoncin, F., Scaillet, O., 2007. Optimal Portfolio Allocation for Pension Funds under Mortality Risk during Accumulation and Decumulation Phases. Annals of Operations Research, 152, 141-165

[5] Muermann, A., Mitchell, O.S, Volkman, J.M., 2006. Regret, Portfolio Choice, and Guarantees in Defined Contribution Schemes. Insurance Mathematics and Economics. 39, 219-229

[6] Braun, M., Muermann, A., 2003. The Impact of Regret on the Demand for Insurance. Research Working Paper Series, The Wharton School University of Pennsylvania, 1-41

[7] Dodonova, A., Khoroshilov, Y., 2005. Applications of Regret Theory to Asset Pricing. Working Paper Series, JEL Classification: G12, G19. Pages 1-30

[8] Heybati, F., Roodposhti, F.R, Moosavi, S.R., 2011. Behavioural Approach to Portfolio Selection: The Case of Tehran Stock Exchange as Emerging Market. African Journal of Business Management, Vol.5, 17, 7593-7602

[9] Battochio, P., Menoncin, F., Scaillet, O., 2003. Optimal Asset Allocation for Pension Funds under Mortality Risk during Accumulation and Decumulation Phases. Research Working Paper Series, JEL Classification: G23, G11, pages 1-16

[11 ] Brennan, M.J., Schwartz, E.S., Lagnado, R., 1997. Strategic Asset Allocation. Journal of Economic Dynamic and Control, 21, 1377-1403

[12] Ait-Sahalia, Y., 1996. Testing Continuous Time Models of the Spot Interest. The Review of Financial Studies, Vol.9, 2, 385-426

[13] Loomes, G., Sugden, R., 1982. Regret Theory: An Alternative Theory of Rational Choice under Uncertainty. The Economic Journal, Vol.92, 36, 805-824.

[14] Ang, A., Bekaert, G., Liu, J., 2005. Why stocks may disappoint. Journal of Financial Economics, 76, 471-508

[15] Camille, N., Coricelli, G., Sallet, J., Pradat-Diehl, P., Duhamel, J.R., Sirigu, A., 2004. The involvement of the orbitofrontal cortex in the experience of regret. Science, 304, 1167-1170.

[16] Coricelli, G., Critchley, H. D., Joffily, M., O’ Doherty, J. P., Sirigu, A., Dolan, R. J., 2005. Regret and its avoidance: a neuroimaging study of choice behavior, Nature Neuroscience, 8, 9, 1255-1262

[17] Golvich, T., Medvic, V.H., 1995. The experience of regret: what, when, and why. Psychology Review, 102, 379-95

[18] Gul, F., 1991. A theory of disappointment aversion. Econometrica, 59(3) 667-86

[19] Kahneman, D., Tversky, A., 1979. Prospect theory: an analysis of decisions under risk. Econometrica, 47, 263-291

[20] Sorum, P. C., Mullet, E., Shim, J., Bonnin-Scaon, S-, Chasseigne, G., Cogneau, J., 2004. Avoidance of anticipated regret: the ordering of prostate-specific antigen tests. Medical Decision Making, 24, 149-159

[21] Zeelenberg, M., van Dijk, W. W., Manstead, A. S. R., vander Plight, J., 2000. On bad decisions and disconfirmed expectancies: the psychology of regret and disappointment. Cognition and Emotion, 14, 521-541

Appendix 1

Derivation of the Feasibility Condition

If 𝑢 represents the constant contribution rate to a pension fund and 𝑣 represents the constant pension

rate paid to a subscriber, then the feasibility condition holds for the pair (𝑢, 𝑣), 𝑢 > 0 and 𝑣 > 0 if

𝔼 [∫𝑚(𝑡)𝑒−𝑟𝑡𝑑𝑡

𝜏

0

] = 0, and the resulting expression for the feasibility condition is

𝑢

𝑣= −1+

1 − 𝔼[𝑒−𝑟𝜏]

1 − 𝔼[𝑒−𝑟𝜏𝕀𝜏<𝑇] − 𝑒𝑟𝑇ℙ(𝜏 ≥ 𝑇)

where 𝜏 is the random death time.

Proof

Given that 𝑇 is the retirement date:

When 𝑡 < 𝑇, we have 𝕀𝑡<𝑇 = 1 and 𝑚(𝑡) = 𝑑𝑈(𝑡)

𝑑𝑡= 𝑢 > 0 ⇒ contributions are made,

money enters the fund and so we are in the accumulation phase.

When 𝑡 ≥ 𝑇, we have 𝕀𝑡<𝑇 = 0 and 𝑚(𝑡) = −𝑑𝑉(𝑡)

𝑑𝑡= −𝑣 < 0 ⇒ pensions are paid, money

leaves the fund and so we are in the decumulation phase.

If we denote the expected present value of all pensions by 𝔼𝑃𝑉𝑃, where

𝔼𝑃𝑉𝑃 = 𝔼∫𝑣(1 − 𝕀𝑡<𝑇)

𝜏

0

𝑒−𝑟𝑡𝑑𝑡

and the expected present value of all contributions by 𝔼𝑃𝑉𝐶, where

𝔼𝑃𝑉𝐶 = 𝔼∫𝑢𝕀𝑡<𝑇

𝜏

0

𝑒−𝑟𝑡𝑑𝑡

then from the point of view of the subscriber, 𝔼𝑃𝑉𝑃 ≥ 𝔼𝑃𝑉𝐶, i.e.

𝐴: 𝔼∫𝑣(1 − 𝕀𝑡<𝑇)

𝜏

0

𝑒−𝑟𝑡𝑑𝑡 ≥ 𝔼∫𝑢𝕀𝑡<𝑇

𝜏

0

𝑒−𝑟𝑡𝑑𝑡

and from the point of view of the pension fund, 𝔼𝑃𝑉𝑃 ≤ 𝔼𝑃𝑉𝐶, i. e

𝐵: 𝔼∫𝑣(1 − 𝕀𝑡<𝑇)

𝜏

0

𝑒−𝑟𝑡𝑑𝑡 ≤ 𝔼∫𝑢𝕀𝑡<𝑇

𝜏

0

𝑒−𝑟𝑡𝑑𝑡

Accordingly, therefore, both parties reach an agreement when 𝐴 ∩ 𝐵 ≠ ∅, i. e.

𝔼∫𝑣(1 − 𝕀𝑡<𝑇)

𝜏

0

𝑒−𝑟𝑡𝑑𝑡 ≥ 𝔼∫𝑢𝕀𝑡<𝑇

𝜏

0

𝑒−𝑟𝑡𝑑𝑡 ∩ 𝔼∫𝑣(1 − 𝕀𝑡<𝑇)

𝜏

0

𝑒−𝑟𝑡𝑑𝑡 ≤ 𝔼∫𝑢𝕀𝑡<𝑇

𝜏

0

𝑒−𝑟𝑡𝑑𝑡 ≠ ∅

⇒ 𝔼∫𝑢𝕀𝑡<𝑇

𝜏

0

𝑒−𝑟𝑡𝑑𝑡 = 𝔼∫𝑣(1 − 𝕀𝑡<𝑇)

𝜏

0

𝑒−𝑟𝑡𝑑𝑡

⇒ 𝔼∫(𝑢𝕀𝑡<𝑇 − 𝑣(1 − 𝕀𝑡<𝑇))

𝜏

0

𝑒−𝑟𝑡𝑑𝑡 = 0, i. e.

𝔼 [∫𝑚(𝑡)𝑒−𝑟𝑡𝑑𝑡

𝜏

0

] = 0,

where 𝑚(𝑡) = 𝑢𝕀𝑡<𝑇 − 𝑣(1 − 𝕀𝑡<𝑇), 𝑒−𝑟𝑡 is the discount factor and 𝜏 is the random death time of

the subscriber. This is the feasibility condition that has to be satisfied before the subscriber and the

pension fund can accept the pair (𝑢, 𝑣). It guarantees that neither the pension fund nor the subscriber

feels cheated.

We next prove

𝑢

𝑣= −1+

1 − 𝔼[𝑒−𝑟𝜏]

1 − 𝔼[𝑒−𝑟𝜏𝕀𝜏<𝑇] − 𝑒𝑟𝑇ℙ(𝜏 ≥ 𝑇)

Eliminating 𝑚(𝑡) from 𝔼[∫ 𝑚(𝑡)𝑒−𝑟𝑡𝑑𝑡𝜏

0] = 0 and 𝑚(𝑡) = 𝑢𝕀𝑡<𝑇 − 𝑣(1 − 𝕀𝑡<𝑇) yields

𝑢

𝑣=𝔼[∫ 𝑒−𝑟𝑡𝑑𝑡

𝜏

0] − 𝔼[∫ 𝕀𝑡<𝑇𝑒

−𝑟𝑡𝑑𝑡𝜏

0]

𝔼[∫ 𝕀𝑡<𝑇𝑒−𝑟𝑡𝑑𝑡𝜏

0]

𝑢

𝑣=

𝔼[∫ 𝑒−𝑟𝑡𝑑𝑡𝜏

0]

𝔼[∫ 𝕀𝑡<𝑇𝑒−𝑟𝑡𝑑𝑡𝜏

0]− 1

Now, since we can write

∫𝑒−𝑟𝑡𝑑𝑡

𝜏

0

=1

𝑟−𝑒−𝑟𝜏

𝑟

and

∫𝕀𝑡<𝑇𝑒−𝑟𝑡𝑑𝑡

𝜏

0

= {

1 − 𝑒−𝑟𝜏

𝑟 for 𝜏 < 𝑇

1 − 𝑒−𝑟𝑇

𝑟 for 𝜏 ≥ 𝑇

then we have

𝔼 [∫ 𝑒−𝑟𝑡𝑑𝑡

𝜏

0

] =1

𝑟−𝔼(𝑒−𝑟𝜏)

𝑟=1

𝑟(1 − 𝔼(𝑒−𝑟𝜏))

and

𝔼 [∫𝕀𝑡<𝑇𝑒−𝑟𝑡𝑑𝑡

𝜏

0

] = ∫1 − 𝑒−𝑟𝜏

𝑟

𝑇

0

𝑓(𝜏) + ∫1− 𝑒−𝑟𝑇

𝑟𝑓(𝜏)

∞

𝑇

where 𝑓(𝜏) is the density of the random death time 𝜏.

Complete integration by parts yields

𝔼 [∫𝕀𝑡<𝑇𝑒−𝑟𝑡𝑑𝑡

𝜏

0

] =1

𝑟(ℙ(𝜏 < 𝑇) + ℙ(𝜏 ≥ 𝑇)) −

1

𝑟∫𝑒−𝑟𝜏𝑓(𝜏)𝑑𝜏 −

1

𝑟

𝑇

0

𝑒−𝑟𝑇ℙ(𝜏 ≥ 𝑇)

or

𝔼 [∫ 𝕀𝑡<𝑇𝑒−𝑟𝑡𝑑𝑡

𝜏

0

] =1

𝑟(1 − 𝔼(𝑒−𝑟𝜏𝕀𝜏<𝑇) − 𝑒

−𝑟𝑇ℙ(𝜏 ≥ 𝑇))

Thus

𝑢

𝑣= −1 +

(1 − 𝔼(𝑒−𝑟𝜏))

1 − 𝔼(𝑒−𝑟𝜏𝕀𝜏<𝑇) − 𝑒−𝑟𝑇ℙ(𝜏 ≥ 𝑇)

as required to prove.

Remarks

The proposition shows that pensions are proportional to contributions. This is exactly what is observed

in practice; the higher the contributions towards retirement, the higher the pensions at retirement.

Appendix 2

Computation of the Feasibility Condition

Battocchio, Menoncin and Scaillet (2003, 2007) assume death time 𝜏 follows a Gompertz-Makeham

distribution and a Weibull distribution and they compute the feasibility condition based on these

assumptions. Here, we explicitly compute the feasibility condition by supposing that the death time 𝜏

follows a log-logistic distribution. Besides giving a concrete view of the feasibility condition, the log-

logistic distribution takes only positive arguments, provides a better characterization of the death time

and is most widely used in death/survival analysis. These are some of its desirable properties which

explain why we favor it in our work.

The log-logistic distribution density function of the death time 𝜏 is given by

𝑓(𝜏) =

𝛽𝛼(𝜏𝛼)𝛽−1

[1 + (𝜏𝛼)

𝛽

]2 where 𝜏 > 0, 𝛼 > 0, 𝛽 > 0

𝛼 and 𝛽 are the scaling and shaping factor respectively.

Figure 1: Log-logistic distribution of death time 𝜏 with 𝛼 = 1.7, 𝛽 = 1.3

The expected time of death is given by

∫ 𝜏𝑓(𝜏)𝑑𝜏 = 𝛼B (1 −1

𝛽, 1 +

1

𝛽) = 𝛼𝛤 (1 −

1

𝛽)𝛤 (1 +

1

𝛽) , 𝛽 > 1

∞

0

where B and 𝛤 are the Beta and Gamma functions respectively. The behavior of the expected death

time is shown in Figure 2 where we have set 𝛼 ∈ [2, 10] and 𝛽 ∈ [1.10, 1.50]. We see that the

expected time of death soars to roughly 100 years when the parameters take the set values above.

Figure 2: Expected time of death for the log-logistic distribution

Before we compute the feasibility condition, we need to first obtain ℙ(𝜏 ≥ 𝑇), 𝔼(𝑒−𝑟𝜏) and

𝔼(𝑒−𝑟𝜏𝕀𝜏<𝑇) under our assumption of a log-logistic death time 𝜏 distribution.

ℙ(𝜏 ≥ 𝑇)

The probability that the death time 𝜏 would occur on or after the obligatory retirement date 𝑇 is given

by

ℙ(𝜏 ≥ 𝑇) =𝛽

𝛼∫

(𝜏𝛼)

𝛽−1

[1 + (𝜏𝛼)

𝛽

]2 𝑑𝜏

∞

𝑇

If we apply the change of variable 𝑢 = 1 + (𝜏

𝛼)𝛽

, where 𝑢 → 0 𝑎𝑠 𝜏 → ∞,we will obtain

ℙ(𝜏 ≥ 𝑇) =1

1 + (𝑇𝛼)

𝛽

We remark that if the retirement date is very far into the future, the probability of death time occurring

after the retirement date tends to zero. This means that the death time is most likely to occur before the

retirement date and thus there is a risk that the employee may die in active service before retirement.

This explains why more countries are increasingly favoring earlier retirement dates.

2

6

10

0

20

40

60

80

100

11.2

1.31.4

1.5

𝛼

𝔼(𝛕)

𝛽

𝔼(𝑒−𝑟𝜏)

This is the expected value of the discounting factor over the death time of the subscriber. An

advantage of the log-logic distribution is that it takes only positive death time, unlike a Normal

distribution which can assume an undesirable negative death time. Considering this, we have

𝔼(𝑒−𝑟𝜏) =𝛽

𝛼∫ 𝑒−𝑟𝜏

∞

0

(𝜏𝛼)

𝛽−1

[1 + (𝜏𝛼)

𝛽

]2 𝑑𝜏 = ∫ 𝑢−2𝑒−𝑟𝛼(𝑢−1)

1𝛽

∞

1

𝑑𝑢

where we have used the change of variable 𝜏 = 𝛼(𝑢 − 1)1

𝛽 . Since this integral does not admit an

elementary algebraic solution, we may propose an approximation as in the Proposition below.

Proposition

Under the assumption that (𝑢 − 1)1

𝛽 < 1 for 𝛽 > 0, 𝔼(𝑒−𝑟𝜏) approximates to

𝔼(𝑒−𝑟𝜏) ≅β

rα[(1

rα)β−1

Γ(β) − 2 (1

rα)2β−1

Γ(2β)]

where Γ is the complete gamma function defined as

Γ(s) = ∫ ts−1

∞

0

e−tdt = (s − 1)!

Proof

Set 𝑡 = 𝑟𝛼(𝑢 − 1)1

𝛽 and use negative binomial theorem to expand [1 + ( 𝑡𝑟𝛼)𝛽

]−2

after which the

result follows.

𝔼(𝑒−𝑟𝜏𝕀𝜏<𝑇)

Computing 𝔼(𝑒−𝑟𝜏𝕀𝜏<𝑇) is akin to computing 𝔼(𝑒−𝑟𝜏) but with an added restriction that the death

time must occur before the retirement date, i.e. given that the subscriber dies before the retirement

date. Arguments similar to Proposition 2 give

𝔼(𝑒−𝑟𝜏𝕀𝜏<𝑇) ≅𝛽

𝑟𝛼[(1

𝑟𝛼)𝛽−1

∫𝑡𝛽−1𝑒−𝑡𝑑𝑡 − 2 (1

𝑟𝛼)2𝛽−1

𝑇

0

∫𝑡2𝛽−1𝑒−𝑡𝑑𝑡

𝑇

0

]

If we define the lower incomplete gamma function as

𝛾(𝑠, 𝑥) = ∫𝑡𝑠−1𝑒−𝑡𝑑𝑡

𝑥

0

then we will have

𝔼(𝑒−𝑟𝜏𝕀𝜏<𝑇) ≅𝛽

𝑟𝛼[(1

𝑟𝛼)𝛽−1

𝛾(𝛽, 𝑇) − 2 (1

𝑟𝛼)2𝛽−1

𝛾(2𝛽, 𝑇) ]

Accordingly, we plug these closed-form approximations for ℙ(𝜏 ≥ 𝑇), 𝔼(𝑒−𝑟𝜏) and 𝔼(𝑒−𝑟𝜏𝕀𝜏<𝑇) into

the feasibility condition to get

𝑢

𝑣≅

𝛽 [(1𝑟𝛼)

𝛽−1

Γ(β, T) − 2 (1𝑟𝛼)

2𝛽−1

Γ(2β, T)] + 𝑟𝛼𝑒−𝑟𝑇

[1 + (𝑇𝛼)

𝛽

]

𝑟𝛼 − 𝛽 [(1𝑟𝛼)

𝛽−1

𝛾(𝛽, 𝑇) − 2 (1𝑟𝛼)

2𝛽−1

𝛾(2𝛽, 𝑇)] − 𝑟𝛼𝑒−𝑟𝑇

[1 + (𝑇𝛼)

𝛽

]

where Γ(𝑠, 𝑥) is the upper incomplete gamma function defined as

Γ(𝑠, 𝑥) = ∫ 𝑡𝑠−1𝑒−𝑡𝑑𝑡

∞

𝑥

and

𝛾(𝑠, 𝑥) + Γ(𝑠, 𝑥) = Γ(s) = ∫ 𝑡𝑠−1𝑒−𝑡𝑑𝑡

∞

0

= (𝑠 − 1)!

This is the condition that has to be satisfied for the pension fund and the subscriber to agree on a

pension contract when the death time of the subscriber is assumed to follow a log-logistic distribution.

We present the results of the feasibility condition for several values of 𝛼, 𝛽, 𝑇 and 𝑟 in Table 1 below.

𝑟 𝛼 𝛽 𝑇 𝑢

𝑣

0.05

0.05

0.05

20

20

20

1.1

1.1

1.1

30

20

10

0.0074

0.0936

0.1322

0.05

0.05

0.05

20

20

20

1.05

1,15

1.20

30

30

30

0.2201

0.0651

0.0602

0.05 10 1.1 30 0.0095

0.05

0.05

15

25

1.1

1.1

30

30

0.0311

0.0721

0.04

0.05

0.06

20

20

20

1.1

1.1

1.1

30

30

30

0.0458

0.0987

0.1245

Table 1: Approximation for the feasible ratio

From Table 1 it is clear that, given the age of a subscriber, when the retirement date 𝑇 increases, the

feasible ratio 𝑢

𝑣 decreases and so the pension fund can afford to pay a higher pension rate to the

subscriber. In fact, the pension fund can demand lower contribution rates when the contributions are

made for a long period of time. Furthermore, when the retirement date 𝑇 is sufficiently far away, the

feasible ratio 𝑢

𝑣 is decreasing with respect to 𝛽 and increasing with respect to 𝛼. Our result contrasts

with Battocchio, Menoncin and Scaillet (2003, 2007) who find a decreasing relationship between 𝑢

𝑣

and both 𝛼 and 𝛽 for a sufficiently large retirement date 𝑇.

Table 1 also shows that the higher the short term interest rate 𝑟 the higher the feasible ratio and

therefore the lower the pension rate the pension fund can afford to pay. In fact, when the interest rate

increases it becomes more difficult for the pension fund to meet future payments. This will

consequently force the pension fund to demand higher contribution rates. Again our result contrasts

sharply with Battocchio, Menoncin and Scaillet (2003, 2007) who find an inverse relationship between

the feasible ratio 𝑢

𝑣 and the short term interest rate. We also find that, as the retirement date 𝑇

increases, the probability that the death time will occur after the retirement date decreases. This means

that more and more, it gets more and more likely that the subscriber will die before retirement or while

in service.

Using the derived approximated ratio, we can graphically depict the behavior of the ratio 𝑢

𝑣 with

respect to the parameters 𝛼 and 𝛽. These graphs are shown in Figure 3, where three different values of

𝑇 and 𝑟 are chosen. In particular, 𝑇 = 10, 20, 30 and 𝑟 = 0.05, 0.06, 0.07 and the values of 𝛼 and 𝛽

belong to [10,20] and [1.05, 1.15] respectively.

Figure 3: Feasible ratio 𝑢

𝑣

𝑇 = 30, 𝑟 = 0.045 𝑇 = 30, 𝑟 = 0.05

𝑇 = 20, 𝑟 = 0.045 𝑇 = 30, 𝑟 = 0.06

10

15

20

0

0.01

0.02

0.03

0.04

𝛼

𝛽

10

17,5

0

0.01

0.02

0.03

0.04

0.05

𝛂

𝛽

10

15

20

0

0.02

0.04

0.06

0.08

0.1

𝛂

𝛽

10

15

20

0

0.01

0.02

0.03

0.04

0.05

𝛂

𝛽

𝑇 = 10, 𝑟 = 0.045 𝑇 = 30, 𝑟 = 0.07

The first column of Figure 3 shows the behavior of the feasible ratio 𝑢

𝑣 for 𝑇 ∈ {10, 20, 30}, while the

second column analyzes how 𝑢

𝑣 changes for 𝑟 ∈ {0.05, 0.06,0.07}. We notice from the second

column of Figure 3 that changes in 𝑟 does not markedly affect the shape of 𝑢

𝑣 . As such, the interest

rate 𝑟 only affects the magnitude or level of 𝑢

𝑣 without concomitantly altering its behavior with

respect to other parameters.

Appendix 3

Proof of Concavity and Monotonicity of the Value Function

Under all the suitably stated conditions that must hold for the optimization problem of the pension

fund, the modified utility function, and hence the value function, is increasing and concave in 𝑊.

Proof

We have

𝜕𝜓

𝜕𝑊= (𝑊(𝑡) − 𝑀(𝑡))

−𝜗(1 + 𝜌 (

1

1 − 𝜗((𝑊(𝑡) −𝑀(𝑡))

1−𝜗− (𝑊𝑜(𝑡) −𝑀(𝑡))

1−𝜗) + 𝑎)

𝜌−1

) > 0

and

𝜕2𝜓

𝜕𝑊2= −𝑄 < 0

10

17,50

0.05

0.1

0.15

0.2

𝛂

𝛽

10

15

20

0

0.01

0.02

0.03

0.04

𝛂

𝛽

where

𝑄 =

[

{

𝜗(𝑊(𝑡) −𝑀(𝑡))

−𝜗−1(1 + 𝜌 (

1

1 − 𝜗((𝑊(𝑡) −𝑀(𝑡))

1−𝜗− (𝑊𝑜(𝑡) − 𝑀(𝑡))

1−𝜗) + 𝑎)

𝜌−1

)

+

𝜌(1 − 𝜌) (1

1 − 𝜗((𝑊(𝑡) − 𝑀(𝑡))

1−𝜗− (𝑊𝑜(𝑡) − 𝑀(𝑡))

1−𝜗) + 𝑎)

𝜌−2

(𝑊(𝑡) − 𝑀(𝑡))−2𝜗

]

> 0

Hence the modified utility is concave in 𝑊, implying that the first order condition is necessary for optimality. This is a standard result in optimization theory. It guarantees that, under suitable conditions for the optimization problem, the value function is increasing and concave in 𝑊. Here the value function is increasing and concave in 𝑊 because it is a function of the modified utility, which is itself increasing and concave in 𝑊.